A large panel needs to be moved through a corridor, the panel is tall as the corridor. The corridor is A feet wide before a right angle turn, after the turn, it is B feet wide. What is the maximum length of the panel that can pass through this corner.

Overhead view of the hallway:

+------------+---
|           /   |
|          /    |B ft
|         /     |
|        /+------
|       / |
|      /  |
|     /   |
|    /    |
|   /     |
|  /      |
| /       |
|/        |
+<-A ft-->|

Two points:

First, it can be longer, so probably it was meant "The shortest it can be..."

And second, a proof that the minimum occurs at 45 degrees is certainly needed..."

If you're talking about the length of the panel, no, we're looking for the maximum length it can be to make the turn.

And as for proof, if you just imagine the set of all triangles, consisting of the N & W wall and a line that intersects them while going through the inside corner, then you'll see that the triangle with the shortest hypotenuse is the one with 45º angles. That would be the longest a panel can be to turn that corner. Obviously anything shorter than that could make it without even touching any of the walls.